Would u please help me with understanding the end of this proof? I want to know that why J=R at the end of the proof. How we could get that? Do we use of “J contains ker(f)” for proving that? How?
Question : Let f:R—>S be a homomorphism of commutative unital rings. Suppose that f is surjective. Prove that if P is a maximal ideal of S then f^(-1)(P) is maximal in R.
Proof: If J is an ideal of R which contains f^(−1)(P) then J contains ker f and f(J) is an ideal of S containing P. Since P is maximal, we have either f(J) = P or f(J) = S. Since J contains the kernel of f, we have J = f^(−1) (P) or J = f^(−1)(S) = R. This proves that f^(−1)(P) is maximal.
